Finite flatness of torsion subschemes of Hilbert-Blumenthal abelian varieties

Abstract

Let E be a totally real number field of degree d over Q. We give a method for constructing a set of Hilbert modular cuspforms f1, . . . , fd with the following property. Let K be the fraction field of a complete dvr A, and let X/K be a Hilbert-Blumenthal abelian variety with multiplicative reduction and real multiplication by the ring of integers of E. Suppose n is an integer such that n divides the minimal valuation of fi(X) for all i. Then X[n′]/K extends to a finite flat group scheme over A, where n′ is a divisor of n with n′/n bounded by a constant depending only on f1, . . . , fd. When E = Q, the theorem reduces to a well-known property of f1 = ∆. In the cases E = Q( √ 2) and E = Q( √ 5), we produce the desired pairs of Hilbert modular forms explicitly and show how they can be used to compute the group of Neron components of a Hilbert-Blumenthal abelian variety with real multiplication by E.